Fractional Calculus · Oscillator Hardware
Athanor Alchemy
Fractional Calculus Oscillator-Based Computers
Hardware that treats oscillation, continuous time, and fractional-order dynamics as native computational primitives — from software emulation to standalone analog chipsets.
A technology company of sNoise Research Laboratory
ATHANOR
Alchemy
Interactive
See the Principle in Motion
Three demonstrations of oscillator-based computation and the Genesis Unity Operator — the transfer function at the core of Fractional Scaling Digital Signal Processing.
1 · Coupled Oscillators
Click or drag to seed oscillators. Nearby nodes couple through a Kuramoto interaction. When the order parameter R rises, the field phase-locks — the physical substrate of the computation.
N=0 · Order R=0.00 incoherent
2 · Phase Relationships & Interference
Two sources with adjustable phase. Constructive and destructive interference is a visual analogy for phase as a computational degree of freedom — the same degree of freedom von Neumann proposed for oscillator-based computers in 1954.
3 · Genesis Unity Operator
Apply H(s) = 1 / sβ/2 in the complex frequency domain. Positive β integrates (memory, smoothing); negative β differentiates (change, roughening). Every magnitude response pivots at the Magnitude Transition Frequency, independent of order.
Gold trace is the filtered output; muted trace is the input. Right panel: log-log magnitude with the MTf pivot marked.
The Principle
From Oscillation to Computation
Oscillator-based computing was first explored in the 1950s. Fractional calculus supplies the mathematical language that makes continuous-time, memory-bearing dynamics practical in modern hardware.
01
Oscillator Foundations
Physical oscillators form a responsive substrate whose phase, amplitude, and coupling can encode and transform information.
02
Fractional-Order Dynamics
Fractional calculus introduces controllable memory and richer continuous-time behavior beyond classical integer-order models.
03
Hardware Realization
The principle is embodied in architectures designed for repeatability, scalability, and direct interaction with physical signals.
04
System Performance
Coupled dynamics deliver measurable capability for demanding computational and signal-processing workloads.
Platform
A Calculus of the Operators
FSDSP operationalizes fractional calculus in the complex Laplace domain. A single operator — the Genesis Unity Operator — performs integration, differentiation, and every fractional order between them by changing one parameter.
GENESIS UNITY OPERATOR
H(s) = [ K / sβ/2 ]æ
- β · scaling exponent
- Sign and magnitude set the order: positive β integrates, negative β differentiates, and β = 0 transmits.
- æ · altitude exponent
- Independent magnitude and phase-tension control across the spectrum, or at a single frequency.
- MTf · unitary pivot
- At f = 1/2π the operator is unity for every β — a stable geometric reference frame.
FSDSP Logic Gates
Continuous-wave equivalents of digital gates, driven by operator inputs (β, æ, K) rather than binary state. Cascaded into a Universal Machine Code.
FOST & SQG
Fractional Orthogonal Superposition interrogates custom frequencies beyond FFT bins. Symmetric Quadrant Generation synthesizes drift-free basis functions.
mFOPID / FSC
Multiplicative fractional-order control in the frequency domain. Proportional, integral, and derivative terms collapse into a Unified Fractional Operator.
HOFE & FCST
Orthogonal fractional encoding and cylindrical spatial multiplexing — stacking independent streams on a single carrier via phase geometry.
Phase-Coherent FC-AI
Deterministic classification by Boolean phase-lock rather than statistical backpropagation. Scales from edge chips to orchestrated expert matrices.
FC-OBC Substrates
Software, FPGA/ASIC, hybrid PCIe accelerators, and standalone analog arrays — LC tanks, memristors, CMOS rings, photonics, spin-torque oscillators.
Applications
What This Enables
Fractional-calculus oscillator-based approaches open new design space across multiple domains — wherever phase, memory, and resonance carry the information.
Artificial Intelligence
Hardware primitives that leverage continuous and fractional-order dynamics for learning and inference — including hallucination-resistant resonant classifiers.
Scientific Computing
Physical modeling and continuous-system problems that benefit from native dynamical hardware rather than integer-order discretization.
Signal & Wave Processing
Complex time-varying signals where phase, memory, and resonance carry primary information — beyond the Heisenberg–Gabor compromise of the FFT.
Control Systems
Fractional-order control architectures with richer memory and gain-invariant iso-damping for flight, robotics, and structural actuation.
Edge & Embedded
Signal-native computation placed closer to sensors and real-world environments, including zero-CPU logic arrays that compute by interference.
Advanced Sensing
Systems that extract structure from noisy or fractional-order natural processes — meteorological, acoustic, biomedical, and kinematic.
Heritage
A Long Arc, Made Practical
In 1954 John von Neumann patented an architecture for computation using the phase relationships of oscillators rather than binary voltage levels (U.S. Patent 2,815,488). Early implementations demonstrated the concept but were constrained by the analog technology of the time — and by the lack of a mathematics that could stabilize nonlinear fractional dynamics without phase ripple.
Decades later, fractional calculus and Fractional Scaling Digital Signal Processing supplied that missing foundation. sNoise Research Laboratory’s issued patents made continuous and fractional-order dynamics computationally tractable at scale, first as digital filters and control systems, then as a universal operator language.
Today
Athanor Alchemy focuses on the hardware realization of fractional-calculus oscillator-based computers — turning the algorithmic foundations developed at sNRL into physical systems.
Visit sNoise Research Laboratory-
1954
von Neumann OBC
Phase of an oscillating signal as a computational primitive.
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2013
Dissertation
Fractional Bode analysis of self-affine natural time series.
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2017–20
FSDSP Patents
Four issued U.S. patents on fractional scaling filters and FOCS.
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2026
Omnibus
Universal Machine Code, FC-OBC, and phase-coherent FC-AI — pending.
Intellectual Property
Issued Foundation
Athanor Alchemy is built on issued U.S. patents held by sNoise Research Laboratory, with an omnibus provisional covering the Universal Machine Code and oscillator-based computing embodiments.
- US 9,740,662 B2 Fractional Scaling Digital Filters and the Generation of Standardized Noise and Synthetic Data Series Aug 22, 2017
- US 10,164,609 B2 Fractional Scaling Digital Signal Processing Dec 25, 2018
- US 10,169,293 B2 Fractional Scaling Digital Filters and the Generation of Standardized Noise and Synthetic Data Series Jan 1, 2019
- US 10,727,813 B2 Fractional Scaling Digital Signal Processing Jul 28, 2020
- Omnibus Provisional Operationalization of Fractional Calculus via FSDSP and FOCS for Universal Machine Code, FC-AI, and FC-OBC May 22, 2026 · pending
Inventor: Jeffrey R. Smigelski, Ph.D. · See the sNRL Patents page for details.
The next computational material
The hardware is the wave.
The next era of computing starts here.
Athanor Alchemy is building the physical systems that make fractional-calculus oscillator-based computation real.